§22.12 Expansions in Other Trigonometric Series and Doubly-Infinite Partial
Fractions: Eisenstein Series

Notes:

For the first right-hand sides of (22.12.2)–(22.12.4)
see Lawden (1989, (8.8.3), (8.8.4), and (8.8.7)). The first right-hand sides
of (22.12.5)–(22.12.13) can be obtained from the first right-hand
sides of (22.12.2)–(22.12.4) by translations of the variable in
accordance with the intersection of the first four rows and first four columns of
Table 22.4.3. For example, in the case of (22.12.5) we apply
the translation cd⁡(z,k)=sn⁡(z+K⁡,k) to (22.12.2). The
second right-hand sides of (22.12.2)–(22.12.13) can be obtained from
the corresponding first right-hand sides by substituting, as appropriate, the expansions
π⁢csc⁡(π⁢ζ)=∑m=-∞∞(-1)m/(ζ-m) or
π⁢cot⁡(π⁢ζ)=limM→∞⁡∑m=-MM1/(ζ-m); compare
(4.22.5) and (4.22.3).
See also Walker (1996, §5.1) and Weil (1999, pp. 14–47).